The dynamics of parallel Schönflies motion generators: The case of a two-limb system

Abstract

The formulation of the mathematical model governing the dynamics of parallel Schönflies motion generators (SMGs) is the subject of this paper. These are robotic systems intended to produce motions that entail displacements of the Schönflies subgroup of rigid-body displacements. Such motions are representative of those produced by the serial robots termed SCARA, which involve three independent translations and one rotation about an axis of fixed direction. The main features of SMGs are illustrated with the aid of the McGill Schönflies motion generator. This robot is composed of two limbs, each being a four-joint RΠΠR - R representing a revolute, Π a Π joint, or a parallelogram linkage - kinematic chain, with only two joints actuated. One important feature of the McGill SMG is its mass distribution, as its moving parts account for about 10 per cent of the mass of its drive units, which are (a) fixed to the robot frame and (b) the only steel parts of the whole robot. Moreover, its joints account for roughly 10 per cent of the total mass of its moving parts, fabricated from aluminium, which justifies neglecting the joint inertia in the mathematical model. This feature calls for a formulation of the model in question in terms of the motor displacements, rather than the joint displacements, as the generalized coordinates of the model. Furthermore, in order to derive the model, the robot is decomposed into two subsystems, the drive and the linkage, the drive being decomposed, in turn, into two subsubsystems, the epicyclic gear train and the right-angled gearbox. The robot kinematics is first derived and then the dynamics model is formulated by means of the natural orthogonal complement. In the framework of this methodology, the inertia and what are called the Coriolis matrices of the mathematical model are additive, in the sense that they can be computed as the sum of the contributions of the different subsystems and subsubsystems of a given mechanical system. The contributions of the subsystems and subsubsystems, in turn, can be computed as the sum of the contributions of the individual moving parts of these. Some general results applicable to all SMGs are derived, which leads to the simplification of the mathematical model of such systems.

Keywords

dynamics of parallel Schönflies motion generators two-limb system mathematical model dynamics model natural orthogonal complement

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References

Dietmaier

‘Inverse kinematics of manipulators with 3 revolute and 3 parallelogram joints.’ In Proceedings of the ASME 22nd Biennial Mechanisms Conference, Scottsdale, 13–16 September 1992, vol. 45, pp. 35–40.

Wohlhart

‘Displacement analysis of the general spatial parallelogram manipulator.’

In Proceedings of the 3rd International Workshop on Advances in robot kinematics, Ferrara, Italy, 1992, pp. 104–111.

Hervé

J. M.

‘The lie group of rigid body displacements, a fundamental tool for mechanism design.’

Mech. Mach. Theory 34 (1999): 719–730

Hartenberg

R. S.

Denavit

Kinematic synthesis of linkages (McGraw-Hill Book Co., New York 1964)

Angeles

‘The qualitative synthesis of parallel manipulators.’

Trans. ASME, J. Mech. Des. 126 (4) (2004): 617–624

Angeles

‘The degree of freedom of parallel robots: A group-theoretic approach.’

In Proceedings of the IEEE International Conference on Robotics and automation, Barcelona, Spain, 18–22 April 2005, pp. 1017–1024.

Clavel

Device for the movement and positioning of an element in space US Patent 4,976,582, 11 December 1990.

Clavel

‘Delta, a fast robot with parallel geometry.’

In Proceedings of the 18th International Symposium on Industrial robots, Lausanne, 1988, pp. 91–100.

Angeles

Morozov

Four-degree-of-freedom parallel manipulator for producing Schönflies motions US Patent 2006/US 7127962 B2, 31 October 2006.

10.

Schönflies

A. M.

‘Über Gruppen von Bewegungen.’

Math. Ann. 28 (1886): 319–342

11.

Schönflies

A. M.

‘Über Gruppen von Transformationen des Raumes in Sich.’

Math. Ann. 34 (1889): 172–203

12.

Hervé

J. M.

‘Analyse structurelle des mécanismes par groupes de déplacements.’

Mech. Mach. Theory 13 (4) (1978): 437–450

13.

Company

Pierrot

Shibukawa

Koji

Four-degree-of-freedom parallel robot Eur. Patent EP1084802, 21 March 2001.

14.

Angeles

Fundamentals of robotic mechanical systems (Springer, New York 2007)

15.

Gauthier

J.-F.

Angeles

Nokleby

Morozov

‘The kinetostatic conditioning of two-limb Schönflies motion generators.’

Trans. ASME, J. Mechanisms and Robotics (2008) (in press)

16.

Golub

G. H.

Van Loan

C. F.

Matrix computations (The Johns Hopkins University Press, Baltimore, Maryland 1989).

17.

Angeles

Lee

‘The formulation of dynamical equations of holonomic mechanical systems using a natural orthogonal complement.’

Trans. ASME, J. Appl. Mechanics 55 (1) (1988): 243–244

18.

Juvinall

Marshek

Fundamentals of machine component design, 3rd edition (John Wiley & Sons, Inc., New York 2000)